I'd need some bounds on the size of Boolean formulas (over $\land$, $\lor$ and $\neg$) computing the multiplication of two integers.
I'm not an expert in circuit complexity and I'm crawling through literature. As far as I know, it's known that multiplication is not in $\mathsf{AC}^0$, i.e. it cannot be computed by a polynomial-size circuit of constant depth, because it can be used to compute Parity.
But, first of all, what if I relax the restriction on depth? Can it be computed by polynomial-size circuits of, say, logarithmic depth? This would still be a polynomial-size circuit as a whole, right?
Then, this is for circuits, but does this tell me anything on the size of Boolean formulas? At worst, a formula can be exponentially larger than the circuit, but does it happen specifically for any formula for multiplication?
In other words, are there polynomial-size formulas to compute integer multiplication?